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Derivative of Logarithmic Functions Calculator

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1

Here, we show you a step-by-step solved example of derivative of logarithmic functions. This solution was automatically generated by our smart calculator:

$\int x\:\ln\:4x\:dx$
2

We can solve the integral $\int x\ln\left(4x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{4x}\frac{d}{dx}\left(4x\right)$

The derivative of the linear function times a constant, is equal to the constant

$4\left(\frac{1}{4x}\right)\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$4\left(\frac{1}{4x}\right)$

Multiplying the fraction by $4$

$\frac{1\cdot 4}{4x}$

Any expression multiplied by $1$ is equal to itself

$\frac{4}{4x}$

Simplify the fraction $\frac{4}{4x}$ by $4$

$\frac{1}{x}$
3

First, identify or choose $u$ and calculate it's derivative, $du$

$\begin{matrix}\displaystyle{u=\ln\left(4x\right)}\\ \displaystyle{du=\frac{1}{x}dx}\end{matrix}$
4

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=xdx}\\ \displaystyle{\int dv=\int xdx}\end{matrix}$
5

Solve the integral to find $v$

$v=\int xdx$
6

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

$\frac{1}{2}x^2$

Multiplying fractions $\frac{1}{x} \times \frac{1}{2}$

$\frac{1}{2}x^2\ln\left|4x\right|-\int\frac{1}{2x}x^2dx$

Multiplying the fraction by $x^2$

$\frac{1}{2}x^2\ln\left|4x\right|-\int\frac{x^2}{2x}dx$

Simplify the fraction $\frac{x^2}{2x}$ by $x$

$\frac{1}{2}x^2\ln\left|4x\right|-\int\frac{x}{2}dx$
7

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{2}x^2\ln\left|4x\right|-\int\frac{x}{2}dx$

Take the constant $\frac{1}{2}$ out of the integral

$- \left(\frac{1}{2}\right)\int xdx$

Multiply the fraction and term in $- \left(\frac{1}{2}\right)\int xdx$

$-\frac{1}{2}\int xdx$

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

$-\frac{1}{2}\cdot \frac{1}{2}x^2$

Multiplying fractions $-\frac{1}{2} \times \frac{1}{2}$

$-\frac{1}{4}x^2$
8

The integral $-\int\frac{x}{2}dx$ results in: $-\frac{1}{4}x^2$

$-\frac{1}{4}x^2$
9

Gather the results of all integrals

$\frac{1}{2}x^2\ln\left|4x\right|-\frac{1}{4}x^2$
10

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{2}x^2\ln\left|4x\right|-\frac{1}{4}x^2+C_0$

Final answer to the problem

$\frac{1}{2}x^2\ln\left|4x\right|-\frac{1}{4}x^2+C_0$

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